Acta Psychologica 129 (2008) 26–31

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Acta Psychologica

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Enumeration: Shape information and expertise

Roy Allen *, Peter McGeorgeThe School of Psychology, College of Life Sciences and Medicine, University of Aberdeen, William Guild Building, Kings College, Aberdeen AB24 2UB, Scotland, United Kingdom

a r t i c l e i n f o

Article history:Received 13 December 2005Received in revised form 28 March 2008Accepted 5 April 2008Available online 20 May 2008

PsycINFO classification:23232346

Keywords:AttentionEnumerationSubitizingExpertisePerceptual groupingMultiple object tracking

0001-6918/$ - see front matter � 2008 Elsevier B.V. Adoi:10.1016/j.actpsy.2008.04.003

* Corresponding author. Tel.: +44 (0)1224 272665;E-mail address: roy.allen@abdn.ac.uk (R. Allen).

a b s t r a c t

This study examined the interaction between grouping information and expertise in a simple enumera-tion task. In two experiments, participants made rapid judgements about the number of items present ina visual display. Within each display, items were grouped into a canonical representation (e.g., triangle,square, and pentagon) or were arranged linearly. In both experiments, grouping information facilitatedenumeration performance, replicating previous findings in the literature. In Experiment 2, the facilitativeeffect of grouping information was found to be greater for Air Traffic Controllers (ATCs) than for matchednovices, though they were no better than novices on linear arrays. This may be because linear, like canon-ical arrays, hold unique numerosity information, but only when they contain the minimum number ofpoints necessary to define a line (i.e., 2). So ATCs’ performance on linear arrays containing more thantwo items does not benefit from a facilitative effect of grouping information. That their experience ofbeing ATCs, in terms of years served, was shown to account for the expertise effect suggests that suchvisuospatial expertise is acquired through frequent exposure to spatial arrays.

� 2008 Elsevier B.V. All rights reserved.

1. Introduction the familiar patterns than to random arrangements. Similar re-

When asked to make a rapid decision about how many itemsare presented in a visual display, people typically show a distinc-tive pattern of response times and accuracy (e.g., Trick & Pylyshyn,1993, 1994). For displays containing small numbers of items (up to3 or 4), accuracy tends to be close to ceiling and response times arefast and relatively constant. For displays containing greater num-bers of items, accuracy generally decreases and response times in-crease as a function of each additional item in the display.Kaufman, Lord, Reese, and Volkmann (1949) coined the term subi-tizing to describe the rapid and accurate enumeration of smallnumbers of items and to distinguish it from the processes of count-ing or estimating involved in quantifying larger (>4) numbers ofitems.

Mandler and Shebo (1982) proposed that subitizing was the re-sult of geometric cues in the arrangement of items in the displayleading to fast pattern recognition and access to associated infor-mation on numerosity (i.e., a triangular pattern is associated withthe number three; a square with the number four, etc.). They pre-sented participants with displays in which items were arrangedeither in a familiar pattern, such as is seen on the face of a die,or randomly. Participants demonstrated a pattern recognitionadvantage in that they responded faster and more accurately to

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sponses were reported by Wender and Rothkegel (2000) who, inaddition, demonstrated that when presented with more complexdisplays, participants would, where possible, partition these intosmall canonical patterns prior to enumeration. In fact, that enu-meration is easier when the elements group in a manner conduciveto form recognition has been extensively explored (see, for exam-ple, van Oeffelen & Vos, 1982; Vos, van Oeffelen, Tibosch, & Allik,1988).

The fact that subitizing appears restricted to displays containingup to about four items has been attributed to the difficulties thatarise when generating canonical patterns for displays of greaternumbers of items. As the number of items within a display in-creases, the number of possible configurations into which theycan be arranged becomes too large to facilitate the developmentof simple representative patterns. For example, Logan and Zbrodoff(2003) demonstrated that perceived similarity between differentconfigurations of the same number of elements decreased as thenumber of elements in the displays increased. Similarity betweendisplays containing three items was very high but then fell dramat-ically as the number of elements per display increased beyond thispoint.

Difficulties in generating a canonical pattern notwithstanding,there is evidence suggesting that the pattern recognition advan-tage can be extended, albeit to a small extent, by practice. Forexample, Mandler and Shebo (1982) demonstrated that, witharound 50 trials using fixed patterns, response times to displays

R. Allen, P. McGeorge / Acta Psychologica 129 (2008) 26–31 27

with more than 4 items fell. This practice effect has also been re-ported by Wolters, Van Kempen, and Wijlhuizen (1987). On eachof the five consecutive days they tested subjects on their abilityto enumerate displays consisting of between 4 and 18 items. Forone group, items in the displays were presented in different ran-dom configurations on each day, and for the other, in consistentpatterns. Practice with the consistent-pattern stimuli led to bothlarge decreases in response times and improvements in accuracy,while only small improvements were found for the random config-urations. In discussing their results, Wolters et al. suggest that animplication of their findings is that, given sufficient experiencewith the possible configurations of items in a display, subitizingshould be possible for any number of items.

Using a multiple-target tracking task, Allen, McGeorge, Pearson,and Milne (2004) examined the ability of radar operators to keeptrack of the locations of sets of randomly moving identical visualtargets. Radar operators were chosen as their work environmentmeans that they are constantly exposed to complex dynamic visualpattern information in which they need to keep track of the loca-tions of many items. These experts in dynamic spatial cognitionwere found to be significantly better than a matched group of nov-ices at target tracking, typically being able to track additional itemseven under conditions of increased workload.

Trick and Pylyshyn (1993) have proposed that a common mech-anism underpins performance on both multiple-target trackingand enumeration tasks. Green and Bavelier (2006) have shown thatexperienced action video game players show significantly betterperformance on both multiple-target tracking and enumerationtasks. Hence, it is possible that the multiple-target tracking advan-tage shown by the radar operators (expert group) in the Allen et al.(2004) study would also be manifested if they were tested on anenumeration task. Further, if the advantage shown by radar oper-ators in multiple-target tracking is ‘‘in some way” related to thepattern recognition processes, then any advantage in an enumera-tion task might be greater where the stimuli consist of regular/canonical patterns, particularly because of our bias towards regularforms (Feldman, 2000). Moreover, previous research has shownthat special configurations (e.g., collinearity, parallelism, convexityversus concavity) often play a role in perceptual grouping andshape perception, even when it is made task-irrelevant (e.g., Feld-man, 1996, 1997; Kukkonen, Foster, Wood, Wagemans, & Van Gool,1996; Wagemans, Lamote, & Van Gool, 1997; Wagemans, VanGool, Lamote, & Foster, 2000).

The suggestion that pattern information may play a role in mul-tiple-target tracking has received some support from work by Yan-tis (1992). Yantis demonstrated that performance on a multiple-target tracking task was improved for participants who were pro-vided with grouping information during the target acquisitionphase, relative to those not provided with this information. Yantisalso noted that the advantage of being provided with groupinginformation was relatively short-lived, something that he attrib-uted to those participants not explicitly provided with groupinginformation discovering their own grouping strategy.

The following study sets out to test whether air traffic control-lers (ATCs), whose role relies on significant expertise in spatial cog-nition, show an extension to the subitizing range relative tomatched participants without this experience, and whether thisis in someway related to more experience, and so a greater abilityto make use of different geometric patterns for items in a visualdisplay. The aim of Experiment 1 is to replicate the previous find-ing of a performance advantage when stimuli are presented ascanonical patterns (e.g., Mandler & Shebo, 1982; Puts & de Weert,1997; Wender & Rothkegel, 2000; Wolters et al., 1987), using thecurrent stimuli and procedure. Experiment 2 then addresses theinfluence of expertise and the interaction of expertise and patterninformation.

2. Experiment 1

2.1. Method

2.1.1. ParticipantsTwenty-seven psychology students (6 males) at the University

of Aberdeen, aged between 18 and 44 (M = 25.3, SD = 7.93), tookpart in the experiment for course credits. All had normal or cor-rected-to-normal vision.

2.1.2. MaterialsAll stimuli were prepared in advance on a computer-aided

drawing package before being converted to bitmaps for presenta-tion. Each stimulus contained a number of identical items (‘‘+”sin bold Times New Roman, subtending a visual angle of approx1.25o at a viewing distance of approximately 57 cm), ranging from1 to 6, whose extent was always the circumference of a notional50 mm diameter circle centred in the middle of the screen. Whenstimuli contained from 3 to 6 items, these were arranged in canon-ical or linear patterns. In line with the notion of a virtual polygonsuggested by Yantis (1992) as facilitating performance in multi-ple-target tracking, particularly where concavities within the poly-gon were avoided, the canonical patterns were all regulargeometric shapes (i.e., triangle, square, pentagon, and hexagon)with equal sides. Linear patterns were lines of ‘‘+”s, always ar-ranged equidistantly along the full length of the notional 50 mmcircle’s diameter. Linear patterns were used, instead of random ar-rays, because the former seemed less able to convey quantitativeinformation by dint of their arrangement, as the latter always pro-vide opportunities for partial grouping and chunking. The variousstimulus configurations are shown in Fig. 1. (Note, that whilstthe co-linearity of parts of the pluses (+’s) might facilitate the per-cept of a square this was not expected to be significant; nor doesthe situation arise from any other arrangement.)

Each trial began as a static frame consisting of a centralisedblack fixation letter ‘o’ subtending a visual angle of 0.42�, on awhite background subtending a visual angle of 21.5�. After a delayof 500 ms, a bitmap was presented, almost immediately replacedby a screen display with the instruction that participants shouldrespond as to how many items the bitmap image had contained.

Trials were displayed using E-prime software (Psychology Soft-ware Tools Inc., Pittsburgh, PA) running on a 350 MHz Pentium IIPC with a 17-in. monitor set to a resolution of 800 � 600 (SVGA)at a viewing distance of approximately 57 cm.

2.1.3. DesignAll participants undertook an enumeration task in which the

stumlus presentation time (17/34/51 ms – multiples of the com-puter’s refresh rate), number of items (1–6) and arrangement(canonical/linear) were systematically manipulated. To minimisepattern learning during the study, several versions of everyarrangement were created, the orientation of each differing by itsdegree of rotation. Order of presentation was completely random-ised by the presenting software. In total, there were 864 trials thattook approximately 45–60 min to complete.

2.1.4. ProcedureIn order to minimise extraneous visual distractors, participants

were tested individually in a darkened room. Each participant wasinstructed that they were to be presented with a series of imagessuch that each contained a number of identical items. For each im-age, once presented, they were simply to indicate, using the nu-meric keys above the QWERTY keys, how many items it hadcontained. They were also cautioned that they might sometimesfeel they had not been shown an image because the presentation

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Fig. 1. Notional 50 mm diameter circle at the centre of stimulus image showing,from left to right, canonical and linear arrangements for, from top to bottom, 3, 4, 5and 6 elements.

% Accuracy by Number of items

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Fig. 2. Participants’ accuracy (% correct) as a function of the number of items in thestimulus.

28 R. Allen, P. McGeorge / Acta Psychologica 129 (2008) 26–31

time had been so short. They were assured that every trial con-tained an image, and that every image contained at least 1 andno more than 7 items, and that they should always give a response,even when they had not consciously registered an image, basingthis on what they felt was the correct number. Participants weremisled as to the maximum number of items per image so as to re-duce the ‘end-effect’ where participants presented with a stimuluscontaining numerous items will simply respond with what theyhave been told is the upper limit of items in the display (Simon,Peterson, Patel, & Sathian, 1998).

2.2. Results

A repeated-measures ANOVA of the accuracy (% correct) data,collapsed across arrangement, was first carried out, with presenta-tion time (17/34/51 ms) and number of items (1–6) as within-sub-jects factors. An a level of .05 was used at all times and, whereverappropriate, the Greenhouse–Geisser correction was applied.

There was a significant main effect of presentation time(F(2,52) = 6.96, MSE = 194.00, p < .01, g2

p = 0.21) in that the longerthe image was presented the more accurate were the participants’responses (17 ms: M = 88.72, SE = 1.34; 34 ms: M = 89.37,SE = 1.07; 51 ms: M = 90.85, SE = 0.96). Post-hoc t-tests, based onthe Bonferroni correction (p < .017 for multiple comparisons),showed that accuracy at 51 ms presentations differed significantlyfrom both 17 ms (t(26) = 3.12, p = .004) and 34 ms (t(26) = 2.75,

p = .011) presentations, though the latter two did not differ signif-icantly from each other (t(26) = 1.25, p = .224). The interaction be-tween presentation time and number of items (F(5.55,144.43) = 2.132, p = .058, g2

p = 0.08) approached significance.The expected significant main effect of number of items was

found (F(1.90, 49.49) = 106.12, MSE = 37817.25, p < .01, g2p = 0.80),

in that accuracy generally decreased as the number of items in-creased (see Fig. 2), and suggested a subitizing/counting boundaryaround 3–4 items.

This was subsequently confirmed by hierarchical regressionanalyses of the accuracy data, collapsed across presentation timeand arrangement, in line with the procedure reported by Cohen,Cohen, West, and Aiken (2003). Subitizing, being a parallel process,was expected to produce a flat performance profile, uninfluencedby the number of items presented. Counting, on the other hand,being serial, should have a significant linear component and, so,decline in a fashion dependent upon the number of items in thedisplay.

Results showed that performance on trials containing from 1to 3 items was best predicted by a constant (B = 99.32,SE = 0.85), with no significant linear or quadratic componentsand was, thus, unaffected by the number of items in the display,suggestive of a single mental process unaffected by cognitiveload. In contrast, the same analysis across trials containing from1 to 4 items was best predicted by a linear model (F(1,106) = 9.17, p < .01), suggesting the onset of a second, cogni-tively-demanding process around 3–4 items. For trials containingfrom 4 to 6 items, a linear (F(1,79) = 96.65, p < .01) model bestpredicted performance. This had a regression line of % accu-racy = 157.703 � 15.278x (where ‘x’ is the number of items).Solving the regression lines for 1–3 and 4–6 items gives anintercept of 3.82 items, signifying the mean transition from par-allel to serial processing.

A final analysis looked at the effect of the intra-stimulus itemarrangement. Performance on stimuli containing just 1 or 2 itemswas excluded from this analysis as no variation in their arrange-ments is possible. Thus, the repeated-measures ANOVA of theaccuracy (% correct) data consisted of time (17/34/51 ms), numberof items (3–6) and arrangement (canonical/linear) as the within-subjects factors. Note that the data were not collapsed over timebecause the initial ANOVA’s interaction between presentation timeand number of items approached significance.

Again, there was a significant main effect of time(F(2,52) = 5.43, MSE = 396.11, p < .01, g2

p = 0.17). Further, therewere also significant main effects of number of items (F(1.97,51.25) = 104.66, MSE = 54915.79, p < .01, g2

p = 0. 80) and arrange-

R. Allen, P. McGeorge / Acta Psychologica 129 (2008) 26–31 29

ment (F(1,26) = 33.36, MSE = 22800.02, p < .01, g2p = 0.56). But,

these two main effects were moderated by a significant interactionof number of items by arrangement (F(1.42, 36.94 = 10.34,MSE = 10921.84, p < .01, g2

p = 0. 28) (see Fig. 3). Post-hoc t-tests,based on the Bonferroni correction (p < .0125 for multiple compar-isons), showed that the arrangement only had a significant impactwhen there were 4 or more items per stimulus, such that accuracyfor linear arrangements was significantly poorer than canonical (3-items: t(26) = 1.95, ns, 4-items: t(26) = 2.90, p = .002; 5-items:t(26) = 5.17, p = .000021, 6-items: t(26) = 4.02, p = .00045) arrange-ments. There were no other significant interactions, suggestingthat time had no significant effect upon either the number of itemsor their arrangement.

Note that the purpose of manipulating the orientation of thestimuli’s elements was only to increase the apparent number ofstimuli so as to reduce the tendency of participants to spontane-ously learn a pattern-recognition strategy whilst they completedthe experiment, although this was more relevant for the secondexperiment, where experts and novices were tested. Number of tri-als in each orientation were, therefore, insufficient for any mean-ingful analysis.

2.3. Discussion

The results showed that increasing the number of elements inthe display has little influence on performance for stimuli con-taining up to about four elements. But once stimuli contain fouror more elements performance deteriorates with each additionalelement. This result is in line with the previous enumerationstudies and suggests that two different processes are in opera-tion, one dominating when the stimuli contain up to four ele-ments (the subitizing range) and the other effective when thestimuli contain a greater number of elements (the countingrange).

Analysis of the ease with which the stimuli could be groupedindicates that, as predicted, performance is poorest for those dis-plays in which the stimuli are most difficult to group into a virtualpolygon (i.e., the linear arrangements). Compared to the lineararrangements, outside of the subitizing range (>4 items), thecanonically arranged stimuli result in better performance. Thisadvantage, shown for stimuli in which the elements could be easilygrouped, replicates the effect reported by previous studies (e.g.,Dehaene & Cohen, 1994; Mandler & Shebo, 1982; Wender & Roth-kegel, 2000; Wolters et al., 1987).

Performance by intra-stimulus arrangement across Number of items

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Fig. 3. Participants’ accuracy (% correct) by item arrangement as a function of thenumber of items in the stimulus.

3. Experiment 2

Experiment 2 sought to investigate whether any expertise ef-fects in enumeration were associated with the advantageous useof grouping information using pattern recognition of canonicalshapes.

3.1. Method

3.1.1. ParticipantsEighteen students (8 females) at the University of Aberdeen

aged between 18 and 27 (M = 20.50, SD = 2.77), and 18 civilianATCs (2 females) employed at Aberdeen airport aged between 26and 56 (M = 39, SD = 8.56) took part. The latter’s experience asATCs ranged between 1 and 37 years (M = 14.83, SD = 11.11). Allhad normal or corrected-to-normal vision. Professional radar oper-ators were chosen because the nature of their work, in monitoringdisplay screens for long periods of time, and tracking multiple tar-gets and responding appropriately, would qualify them as expertsas regards an MOT task, something together with subitizingthought to be underpinned by visual indexing (Trick & Pylyshyn,1993). Undergraduate students were selected as being representa-tive of the non-expert, or novice, population at large.

3.1.2. Design and procedureThe design and procedure were the same as in Experiment 1

with the exception that hardware changes resulted in differentstimulus presentation times (13 and 26 ms), and the presentationsoftware this time was Superlab Pro (Cedrus Corporation, San Ped-ro, CA).

3.2. Results

A repeated-measures ANOVA of the accuracy (% correct) data,collapsed across arrangement, was first carried out, with presenta-tion time (13/26 ms) and number of items (1–6) as within-subjectsfactors, and expertise (expert/novice) as the between-subjects fac-tor, to identify the subitizing/counting boundary. An alpha level of.05 was again used at all times and, where appropriate, the Green-house–Geisser correction was applied.

There was a significant main effect of presentation time(F(1,34) = 802.22, MSE = 27302.71, p < .01, g2

p = 0.96) in that thelonger the image was presented the more accurate were partici-pants’ responses (13 ms: M = 76.50, SD = 4.48; 26 ms: M = 92.40,SD = 4.47). More critically, there was also a significant interactionbetween presentation time and number of items (F(3.84,130.52) = 22.73, p < .01, g2

p = 0.40). The large difference in meanperformance for 13 and 26 ms presentations, and the significantinteraction between presentation time and number of items (seeFig. 4) suggest that participants are unable to subitize on trials pre-sented for just 13 ms and are always simply estimating.

Hierarchical regression analyses (Cohen et al., 2003) of the13 ms presentation time accuracy (% correct) data, collapsed acrossarrangement, and carried out for novices and experts separately,confirmed that, at such a short presentation time, all participants’were unable to subitize, since there were no significant quadraticcomponents, only significant linear components across the fullrange of item numbers (novice: (F(1,106) = 216.05, p < .01); ex-perts (F(1,106) = 194.16, p < .01)) (i.e., no evidence for a boundarybetween two different processes).

Because subitizing, a key variable to be investigated, was notevidenced with 13 ms presentation times, all subsequent analyseshave been upon 26 ms presentation time accuracy data only. (Notethat since, in Experiment 1, subitizing was evidenced with 17 mspresentation times, its failure to occur at 13 ms presentation times

%Accuracy by presentation time across Number of items

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Fig. 4. Participants’ accuracy (% correct) as a function of the number of items in thestimulus, across presentation times.

30 R. Allen, P. McGeorge / Acta Psychologica 129 (2008) 26–31

may reflect a lower presentation time threshold for subitizingwithin this paradigm).

Results of the repeated-measures ANOVA, for 26 ms presenta-tion times only, revealed the expected significant main effect ofnumber of items (F(1.68, 57.19) = 73.00, MSE = 11839.56, p < .01,g2

p = 0. 68), in that accuracy generally decreased as the number ofitems increased. In addition, there was a main effect of expertise(F(1,34) = 6.30, MSE = 756.20, p < .05, g2

p = 0. 16) in that experts’accuracy (%) (M = 94.27, SD = 6.32) was significantly greater thanthat of novices (M = 90.53, SD = 6.32). An interaction between thenumber of items and expertise (F(1.68, 57.19) = 2.82,MSE = 457.11, p = .077, g2

p = 0. 077) also approached significance(see Fig. 5).

Results of the hierarchical regression analyses for the 26 mspresentation time showed that novices’ performance on trials con-taining from 1 to 3 items was best predicted by a constant(B = 99.15, SE = 0.73). In contrast, an analysis across trials contain-ing from 1 to 4 items produced significant linear (F(1,70) = 13.39,p < .01), and quadratic (F(2,69) = 13.45, p < .01) models, suggesting,much as in experiment 1, the onset of a second, cognitively-demanding process around 3–4 items. For trials containing from4 to 6 items, a significant linear model (F(1,52) = 36.50, p < .01)best predicted performance. This had a regression line of % accu-

Performance by Expertise across Number of items

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Fig. 5. Participants’ accuracy (% correct) as a function of the number of items in thestimulus, across levels of expertise.

racy = 148.98 � 13.368x. Solving the two regression lines givesan intercept of 3.73 items, being novices’ mean transition from par-allel to serial processing.

Experts’ performance on trials containing from 1 to 3 items wasbest predicted by a constant (B = 99.65, SE = 0.43). In contrast, ananalysis across trials containing from 1 to 4 items produced a lin-ear (F(1,70) = 3.31, p = .073) that approached significance againsuggesting a second, cognitively-demanding process onset around3 to 4 items. For trials containing from 4 to 6 items, a significantlinear model (F(1,52) = 44.04, p < .01) best predicted performance.This had a regression line of % accuracy = 138.05 � 9.81x. Solvingthe two regression lines gives an intercept of 3.91 items, being ex-perts’ mean transition from parallel to serial processing. (Note that,whilst experts’ transition point is greater than that for novices,whether this is a significant difference cannot be determined withthis dataset.)

A final ANOVA/ANCOVA comparison analysis sought to localisethe significant expertise effect reported in the initial repeated-measures ANOVA, and to determine what, if any, contribution ex-perts’ years of experience as ATCs had made to this. Since arrange-ment was not significant for 3-item trials for either novices(t(17) < 1) or experts (t(17) < 1), data were simply collapsed across1–3 items, whilst for 4–6-item canonical and linear arrangementperformance was collapsed separately. A univariate ANOVA ofthe collapsed 1–3-item data, with expert as the between-subjectsfactor, showed no significant effect of expertise (F(1,34) = 2.67,MSE = 3.43, p = .11). For the collapsed linear 4–6-item data too,there was no significant effect of expertise (F(1,34) = 2.16,MSE = 458.48, p = .15). However, for the collapsed canonical 4–6-item data, there was a significant effect of expertise(F(1,34) = 7.50, MSE = 391.71, p < .01, g2

p = 0. 18) in that experts(M = 97.69, SD = 1.70) performed significantly better than novices(M = 91.09, SD = 1.70). Further, when experience, in terms of yearsas an ATC, was entered into the ANOVA as a covariate, the effect ofexpertise became marginal (F(1,33) = 4.11, MSE = 221.18, p = .051,g2

p = 0.11) suggesting that such experience accounts for much of theexpertise effect.

3.3. Discussion

The results of Experiment 2 replicate those observed in Experi-ment 1, and confirm previous findings (e.g., Mandler & Shebo,1982; Wender & Rothkegel, 2000; Wolters et al., 1987) that provid-ing participants with grouping information facilitates enumera-tion. The facilitative influence of grouping information is clearlyobservable for arrays consisting of four or more items. For arraysconsisting of fewer than four items performance is not significantlydifferent, reflecting the fact that for these small array sizes there islittle, if any, difference in what constitutes canonical or lineararrays.

Experiment 2 also demonstrated the expected expertise effect.Experts’ response accuracy was significantly better than Novices’.However, this expertise effect was only observable for arrays con-taining more than four items, and then only when presented in acanonical form. That ATCs, whose role consists primarily in themonitoring of spatial configurations, outperform those lackingsuch experience (Novices) is in line with previous research that in-creased exposure to spatial arrays facilitates improvements in enu-meration performance (e.g., Mandler & Shebo, 1982; Wolters et al.,1987). Wolters et al. presented participants with arrays consistingof random patterns of between 4 and 18 dots. These arrays werepresented for five consecutive days of training and testing. Therewere two conditions, in the first the patterns remained the sameduring training and, in the second, different random patterns wereassociated with each numerosity across the training period. Re-sponse times and accuracy improved markedly across training in

R. Allen, P. McGeorge / Acta Psychologica 129 (2008) 26–31 31

the consistent condition but not in the variable condition. Based onthis, Wolters et al. argued that subitizing is the result of patternrecognition and with sufficient practice subitizing would be ob-served for all numerosities. However, as they note, the number ofpatterns that can be associated with a particular numerosity in-creases rapidly once an array includes more than three items, mak-ing it increasingly difficult to employ this strategy for highernumerosities. [It is worth noting that even under the varied condi-tion, there were some improvements in performance.]

The assumption made in the current study is that the everydayrole of ATCs exposes them to a multitude of spatial configurationsand that this experience then facilitates their performance on theenumeration task. This only becomes apparent for higher numer-osities as both ATCs and Novices are close to ceiling performanceat lower numerosities. However, this then raises the question asto why, if ATCs are exposed to a plethora of spatial configurations,does this not also extend to linear arrays. One explanation, respon-sible for the selection of linear arrays as a baseline in the currentstudy, relates to what constitutes featural invariance. Wolterset al. have pointed out that we have no clear insight into what ‘‘fea-tural invariances” underlie our concepts of numerosity (e.g., three-ness, fourness, etc.). What is clear from the current results iswhatever these are they are lacking in linear arrays. The definingdifference between linear and non-linear arrays is that the former’sshape confounds its inherent numerosity when the line is definedby more than two objects. In other words, typically, a straight linewould be associated with the concept of twoness, as a triangle iswith threeness, etc.

A second, but not mutually exclusive, explanation for the lack ofan observable expertise effect for linear arrays might be that linearpatterns are not legitimate patterns, that is, they would not formpart of the everyday experience of an ATC. This line of explanationis similar to that employed to explain the expertise effects shownwhen Expert and Novice chess players recall the positions of chesspieces (e.g., Chase & Simon, 1973). Classically, expert chess playersshow significantly better memory for legitimate patterns of chesspieces than do novice players, but perform at a near equivalent le-vel on random patterns (non-legal patterns). Perhaps having a lin-ear arrangement of aircraft is something that should be avoidedand so there would not be the opportunity to learn this type of con-figuration and associated numerosities.

The results of the current study also have implications forprevious research on multiple-object tracking (e.g., Allen,McGeorge, Pearson, & Milne, 2006; Allen et al., 2004); particu-larly as this task is thought to share an underlying relationshipwith enumeration (see Green & Bavelier, 2006; Trick & Pylyshyn,1993). Yantis (1992) has suggested that multiple-object trackingmay be accomplished through the formation of a virtual polygon,whose apices are defined by the individual items to be tracked,and the subsequent tracking of this object as it alters its shapeas the targets move. Central to Yantis’ argument is that groupingplays a key role in the initial stages of multiple-object tracking.Allen et al.’s demonstration that radar operators are better thannovices at tracking moving targets, together with the present

finding, in which experts are better at using grouping informa-tion in the enumeration task, could be taken as support for thehypothesis that the advantage shown by radar operators in mul-tiple-object tracking may indeed be facilitated by more efficientgrouping strategies.

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